Differential Equations (AN5)

Graduate Courses Outlines - Department of Mathematics

General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AN5
Semester	2
Course Title	Differential Equations
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

This course is aiming at familiarizing the students with a variety of advanced subjects related to differential equations. Both classical and modern subjects are studied. After attending this course, the students should be able to: be familiar with a vast set of subjects related to differential equations, start researching on subjects regarding the qualitative theory of differential equations, and familiarize himself with the bibliography related to the subjects he was taught.

General Competences

Working independently. Team work. Production of new research ideas. Production of free, creative and inductive thinking. Search for, analysis and synthesis of data and information, with the use of the necessary technology. Develop critical thinking skills.

Syllabus

Second order linear ordinary differential equations: Sturm-type theorems, oscillation and non-oscillation theorems. Reducing differential equation problems to integral ones. Volterra integral equations: existence and uniqueness of solutions. Existence of solutions. The linear equation. The first order linear equation. Some problems on the semi-axis. Fredholm theory for linear integral equations: the resolvent kernel. The entire functions of Fredholm and their applications. Eigenvalues, eigenfunctions and conjugate functions. Some integral inequalities: Gronwall and Bihari Lemmas, and their applications. Delay differential equations: Introduction, Examples and the stepping method. Some remarkable examples and some "wrong" questions. Lipschitz condition and uniqueness for the basic initial problem. Notations and uniqueness for systems with bounded delay. Existence for systems with bounded delay. Linear delay differential systems: superposition. Fixed coefficients. Variation of parameters. Stability for delayed differential systems: Definitions and examples. Asymptotic stability. Linear and almost linear differential systems. Fractional differential equations: Definitions and basic calculus. Initial and boundary systems. Dynamical systems: definitions and calculus. Equations and problems. Various subjects.

Teaching and Learning Methods - Evaluation

Delivery	Lectures on class
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Exercises / Homework 52.5 Autonomous Study 96 Course total 187.5
Student Performance Evaluation	The students can choose one of the following options: Presentation in the class - Written homework - Exercises. Written final exam. If both methods are used, then the final grade is the maximum of the two. The criteria regarding the grading are publised in the "E-Course" platform.

Attached Bibliography

• C. Corduneanu, Principles of Differential and Integral Equations
• R. D. Driver, Ordinary and Delay Differential Equations
• T. A. Burton, Volterra Integral and Differential Equations
• R. K. Miller, Nonlinear Volterra Integral Equations
• P. Hartman, Ordinary Differential Equations
• Κ. Diethelm, The Analysis of Fractional Differential Equations
• Y. Zhou, Basic Theory of Fractional Differential Equations
• M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications.